stochastic processes and their

ELSEVIER Stochastic Processes and their Applications 60 (1995) 49-63 applications

On the asymptotic independence of the sum and rare values of weakly dependent stationary random variables

Tailen Hsing ’ Department of Statistics, Texas A & M University, College Station, TX 77843-3143, USA

Received 30 November 1993; revised 30 April 1995

Abstract

It is shown that if the stationary sequence {Xi} has finite variance and satisfies a certain mixing condition, then the asymptotic distribution of c:=, X. is unaffected by the information of whether the sunnnands are in certain “rare” sets. An application of the result shows that c:=, X, and the extremes of XI , . . . ,X,, are asymptotically independent. This is in sharp contrast to the infinite variance case.

AMS 1990 Subject Classifications: Primary 60FOS; secondary 60GlO.

Keywords: Central limit theorem; Extreme value; Mixing condition; Point process

1. Introduction

The joint behavior of the sum and extreme order statistics of random variables in a large sample has been investigated from many different angles. The topic is of interest for good reasons. From a general theoretical point of view, truncation arguments are invariably involved when proving limit theorems for sums of random variables (e.g. laws of large numbers, central limit theorems, domain of attraction theories, etc.), in which context assessing the effect of truncation is essential. In particular, in the heavy-tailed case, the interplay between the sum and extremes is intriguing and is the topic of an array of results in Feller (1971) and some of the references cited in the paragraph below. From a more practical point of view, for instance, in the analysis of environmental data such as temperature, pollution level, water level, and so on, the data recorded are often means and extreme values. It is then a natural question to ask in what way the information contained in the sums and extremes can be combined for a particular data analysis problem (cf. Turkman and Anderson, 1993).

Consider a weakly dependent sequence of random variables {Xi}. Let F(x) = P[X, <xl and 8, = EYE, Xi. In the infinite variance case, the extreme order statis- tics of a large sample Xl,..., X,, dominate S,, therefore it is possible to describe the

’ Research supported by NAVY-ONR Grant N00014-92-J-1007 and the Alexander von Humboldt Foundation.

0304-4149/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 0304-4149(95)00054-2

50 T. HsinglStochastic Processes and their Applications 60 (1995) 49-63

asymptotic distribution of S,, in terms of those of the extreme order statistics. How- e;rer, as a referee points out, it is possible for the upper and lower extremes to cancel one another in the limit so that the sum is asymptotically independent of the extremes. For some specifics of these ideas, see LePage et al. (1981) and Davis and Hsing (1995). On the other hand, if EX: < 00 and S, satisfies the central limit theorem, then the individual summands are asymptotically negligible and hence the asymptotic distribution of S, ought not depend on the extremes. A number of authors attempted to substantiate that intuition. Chow and Teugels (1978) proved the asymptotic inde- pendence of the sum and maximum, assuming that {Xi} is i.i.d., and F is in both the sum-domain of attraction of the normal distribution and the max-domain of attraction of an extreme value distribution. Anderson and Turkman (1991a,b) extended Chow and Teugels’ result by considering a strongly mixing stationary {Xi} (cf. McCormick and Sun, 1993). Ho and Hsing ( 1994) studied the asymptotic joint distribution for the Gaussian sequence. The following result was obtained in Hsing ( 1995).

Theorem 1.1. Let {Xi} be a strictly stationary strongly mixing sequence of random variables with zero mean and finite variance. Assume that oi := ES,” -+ 00 and &/a, converges weakly in distribution to a standard normal random variable. Then for any sequence B, of Bore1 sets in (--00, co) such that

1iminfP h(X E B,) > 0, n-C.3

I 1 i=l

(1.1)

the conditional distribution F,,(x) := P[S,/a, 6x1 n:=l(Xi E B,)] converges weakly to the standard normal distribution.

The most obvious application of Theorem 1.1 is taking B, = (-00, u,,(y)] for some monotone increasing function un(y) for which

1 2 G(y),

where G is a probability distribution. This generalizes the above mentioned result in Anderson and Turkman ( 1991a,b) by removing some conditions that are not essential. It is clear that the generalization also opens doors to other useful applications concern- ing the asymptotic distribution of the sum and rare occurrences of weakly dependent random variables.

While Theorem 1.1 is based on a rather general formulation, it still fails to fully explain whether or why sums and extremes are in general asymptotically independent in the finite variance case. The purpose of the present paper is to explore that and related issues.

The main results are stated in Section 2, whereas the technical details are collected in Section 3. Theorem 2.1 in Section 2 states that if E/X, 12+q < 00 for some q > 0 and {Xi} satisfies a certain mixing condition, then the asymptotic distribution of S, is unchanged given the information of whether the summands are in certain “rare” sets. The proof is largely based on an application of Theorem 1.1 and a detailed analysis of

T. HsinglStochastic Processes and their Applications 60 (1995) 4963 51

how rare events are positioned in a sequence of weakly dependent events. An immediate application of Theorem 2.1 results in Theorem 2.2, which concludes that S, and the

bivariate point process

are asymptotically independent, where 6, is a point mass at x and F = 1 - F. For a weakly dependent sequence, the asymptotic distributions of these point processes capture those of the extreme order statistics. Thus, a clear understanding of asymptotic independence between the sum and the extremes is established.

2. Main results

Let {Xi} be a strictly stationary sequence of random variables. Since the standard mixing conditions are not designed for the purpose of the topic in this paper, it is possible to approach the problem of modeling dependence from a number of different directions. To minimize technicalities in that regard, we will work under the strong mixing condition. To be specific, we first define the mixing coefficient in terms of which the strong mixing condition is defined. For any set I, let P(Z) denote the a-field generated by Xi,j E I. Define, for 1 = 1,2,. . . ,

a(l) = sup{IP[AnB]-P[A]P[B]I :A E 9((-00,0]),B E 9([E,co))}.

Write 8, = (X t,. . . ,X,). With I? = (Br,. . . ,I?,) denoting a sequence of Bore1 sets

Bl,..., B,, the notation 2, E h stands for Xi E Bi, for all i = 1,. . . , n; accordingly, 2, 6 8 if and only if Xi #B; for some i = l,.. . , n. In Theorem 2.1 below, let r be a fixed positive integer and for each n, let B,,i, 1 d i<r, be disjoint Bore1 sets such that

I

U B,,i = R. i=l

Also, @ stands for the standard normal distribution function. We state the main result of this paper as follows.

Theorem 2.1. Let {xi} be strictly stationary with E& = 0 and EIXl12+q < DC) for some q > 0, and with limn+oo nka(n) = 0 for all k E (O,a). Assume additionally that (r2 > 0, where a2 := EXf +2 c:, E(Xl X, .) which exists Jinite by the assumptions stated in the previous sentence, and that the Bore1 sets B,,i are such that for some z E {l,...,Y},

(2.1)

and, with m, := nv/(2(3+‘f)),

fi(& $Z B,,) 1 Xl g! B,, = 0. i=I I

(2.2)

52 T HsinglStochastic Processes and their Applications 60 (1995) 4943

Then there exist sets .~#,,,n > 1, such that for each n, 9#,, contains sequences of Bore1 sets i = (Bl, . . . , B,,) where each Bi E {B,,, , B,,J, . . . B,,}, 1 <i <n, and such that

(2.3)

and

lim sup IP[S,/(ofi) <x ) if, E B] - @(x)l = 0. (2.4) n-C% --oo<X<co, dCd,

Remark 1. For i? = (B I,. . . , B,) whose components are identically equal to B,,, in Theorem 2.1, (2.1) and the strong-mixing condition together imply that

liminfP[f,Eh]=liminfP n-w n-CC (2.5)

(i.e., (1.1) holds with B,, = B,,; see Leadbetter et al., 1983) and hence Theorem 1.1 readily implies that

For large n, while &,, necessarily contains this particular sequence i by (2.3) and (2.5), except for the trivial case where the limit in (2.5) equals 1, @,, does contain other sequences as well. Consequently, the real purpose of (2.4) is to address the case where B contains at least one component different from B,,,, in which case we have,

by (2.1),

P[k, E 81 <P[& +z’ B,,,] ---f 0.

Thus, the asymptotic independence problem in this situation assumes a different char- acter than the one addressed in Theorem 1.1.

Remark 2. The fact that a uniform rate of convergence can be achieved for all se- quences in .@,, in (2.4) may seem a little surprising and can be intuitively explained by the following. Consider the case where BJ = a,x + b, and Bn,2 = Bi,, where a; ’ (max t <i<n Xi - b, ) converges in distribution to some non-degenerate limit. We can think of (Xi E BQ) as observing an extreme value. Under weak dependence, it is well known that the extreme values occur in clusters (cf. Hsing, 1993). Making crucial use of the assumed rate of decay of the mixing coefficient, it can be seen that the clusters of extremes are, in some sense, uniformly asymptotically independent of one another, and of the non-extremes.

Remark 3. Condition (2.2) is also a weak dependence restriction. To see this, note that by Boole’s inequality and (2.1), the unconditional probability

P[g(& i/B..)] <m,P[Xl $z’B,,] --+O for all fixed 1.

T. Hsing I Stochastic Processes and their Applications 60 (1995) 4963 53

Hence, (2.2) requires that the effect of the rare event (XI 6 &) on bringing about another rare event Uy!,(Xi 61 B,,) to die out as I increases.

Remark 4. By the discussions on p. 164 of Chung (1974), (2.4) is equivalent to

lim sup (1 + t*)-‘]&(t]B) - e-r2/21 = 0, (2.4’) n-CC --OO<t<cu, dGl,

where

&(tlE) := E[exp(it&/(ofi))Z(JZ, E E)]

P[2, E B] ’ -ccottco,

i.e., the characteristic function of probability measure P[S,/(ofi) E . 12, E L?]. In fact, we will prove (2.4) through (2.4’), since the characteristic function approach is technically more natural for the strong mixing condition assumed.

Theorem 2.1 answers many questions about the asymptotic independence of S,, and the extremes of X 1, . . ..X.. As an illustration, we consider a straightforward application of Theorem 2.1. Define the following point processes on the state space [O,oc):

n n

NcL) = n c &F(x) and NC”) = n c 6 - n&K )’

i=l i=l

where S, is a point mass at x. Regard these point processes as random elements in k!, the space of locally finite counting measures on [O,oo) equipped with the Bore1 a-field determined by the vague topology. See Kallenberg (1983) for the general theory of random measures. Clearly, the asymptotic distributions of, respectively, the lower and upper extreme order statistics can be derived from those of NiL’ and Nn(“). In the i.i.d. case, if

l im F(x) -= X-XI F(x-)

1 and lim -!(‘) -= 1 X’& F(x-) ’

where xi and x, are, respectively, the left and right end points of the support of F, then NcL) and N(“) converge in distribution to independent homogeneous Poisson processes. In” general, lhese point processes may not converge in distribution, and when they do, the limits may be dependent. See Leadbetter et al. (1983), Hsing (1993), and Davis and Hsing (1995). In particular, Davis and Hsing (1995) showed that in the infinite variance case, the asymptotic distributional behavior of S, depends highly on that of (N,$“‘, A$“‘). The following result provides a contrast in the finite variance case.

Theorem 2.2. Let {xl} be strictly stationary with EX, = 0 and E(X,l*+q < cc for some r] > 0, and with lim,,, nka(n) = 0 for all k E (0, co). Assume additionally that o* > 0, where CT* := EX: +2 CE, E(XlXi) which exists finite by the assumptions stated in the previous sentence, and that

m,

/iii 1imEp P U (nF(Xi) A nF(Xi) <a) ) nF(X, ) A rS(X~ ) f a I

= 0 (2.6) i=l

54 T. Hsing IStochastic Processes and their Applications 60 (1995) 49-63

for all a > 0, where m, := nn/(2(3+n)! Then for any injinite subsequence of the set of positive integers, there exists a further injinite subsequence along which (&/(a,/%), NiL’, N,(“)) converges weakly in the product space (If& A, A) to some random element (2, NcL), NC”)) where Z N @ is independent of (NcL), NC”)).

The proofs of Theorems 2.1 and 2.2 are given in Section 3.

3. Proofs

3.1. Proof of Theorem 2.1

We begin by noting that the assumptions EIXi 12+q < co and a(l) tending to 0 faster than any polynomial rate imply that 0 2 := EX: + 2 cz, E(X&) exists finite, and if d > 0, then the distribution of $/(oJ) n converges weakly to the standard normal distribution. See Section 18.5 of Ibragimov and Linnik (1971). For convenience, let on = r~Ji; henceforth. Also notice that the same assumptions require that

m,nP[IXl( > xn’i2/mn] + 0 as n --f cc for all x > 0 (3.1)

and

n’m(m,) + 0 as n -+ cc for all j > 0,

where m, = n’l/(2(3+q)).

(3.2)

In view of the discussions in Remark 1, it is plausible to call the B,,i, i # z, rare sets for Xi. Let Bl,B2, . . . , B, be n Bore1 sets each equal to one of B,,I , B,,,J, . . . , B,,, but not identically to B,,. For this proof it is convenient to speak about the clusters of rare sets of size m, in the sequence 8 = (B 1, . . . . B,), m,-clusters for short, which we now define. The m,-clusters of B are sets of indices in { 1,2,. . . , n}, where the first m,-cluster starts with

tL (‘) = min(j : 1 <j<n, Bj # B,,,),

which is followed by those indices j with tf ’ + 1 d j <(tf ’ + m, - 1) A n and such that Bj # B,,. If tf ‘, the last element of the first m,-cluster, happens to be max(j : 1 <j<n, Bj # B,,), then there are no further m,-clusters. Otherwise, the second m,- cluster starts with

tL (2) = min(j : tr’+ 1 <j<n,Bj #BE,,),

and followed by the indices j with tf’ + 1 <j < (tf’ + m, - 1) A n and such that

Bj # Bn,r. Further m,-clusters are defined in this manner recursively. The length of a cluster is defined to be l+ (largest element of the cluster - smallest element of the cluster), and the distance between one cluster and the next is defined to be the number of integers strictly between the two clusters. For each m,-cluster C, define the neighborhoods,

C={j: lV(inf(C)-m,)<j<(sup(C)+m,)An,j@C}

T. Hsingl Stochastic Processes and their Applications 60

and

C={j: lV(inf(C)-m,)<j<(sup(Z’)+m,)An,j$

(1995) 49-63

CUC}.

55

IfB r,. . . , B, are all equal to B,,, then there is obviously no m,-clusters to speak about, and we say the number of m,-clusters is 0.

Let

&z,all = {j = (B I,.. .,B,) : each Bi equals one of BJ,..., B,,}.

Since for each n, BJ . . . , B,,, form a disjoint partition of [w,

p[JJ*,MQ] =BE,,,P[~.tB1=l.

For n, k 2 1 and E > 0, define d,,(k,&) to be the collection of all B = (B,, . . . , B,) E dn+ll satisfying the following (i)-(iii):

(i) j has at most k m,-clusters; (ii) the minimum distance between the m,-clusters of B is at least 5m,; and

(iii) for each m,-cluster C in b,

nP n(& E Bi), n(xI E &,T) [ iEC iEC 1 > 8. (3.3)

First, fix ko and ~0. Our first goal is to first show that (2.4’) holds for an = &,,(kc,sc). Consider any sequence B = (B 1, . . . ,B,) E .d:,(ko,@). Suppose j has k m,- clusters, 1 <k <k,, which are denoted by Cl, . . . , Ck. Note that the sequence fi which contains all B,,,‘s has already been dealt with in Theorem 1.1 (cf. Remark 1). Also, let

k

D = {l,..., n} - U(CI u C’I u 6,).

Since each of Cl U cr U e’~ is an interval containing at least 4m, + 1 integers, D is the union of at most k + 1 intervals, denoted by D1, D2, . . . , which are separated from one another by at least 4m, + 1 integers. Also let

T,, = C Xj and T,, = C T,,,. jED, r

Define the events

&I= [Gy EBj)] n [$yg&)] forall 1.

By (3.3) with E = cc,

P[E,,l] > E&Z, for all 1. (3.4)

(3.5)

56 T. Hsinglstochastic Processes and their Applications 60 (1995) 4963

In the rest of the proof, i stands for J-i whenever it is a part of the argument of exp( .). Write

E[exp(iLS,/a,)I(J?, E ii)] - eCt2/*P[kn E 81 = $Anj,

j=l

where

A,1 = E[exp(itS,/a,)Z(*, E l?)]

T HsinglStochastic Processes and their Applications 60 (1995) 49-63 51

rfl 1 k

J n P[E,,,] - e-f2’2P[~n E b]. I=1

We will show that uniformly for g E dn(ko,~g) and t E any fixed compact interval,

1 fi Wn,I Li=l 1 I=I

and

i=l 7. ,...,

(3.6)

(3.7)

Clearly, these imply (2.4’) with d, = J?.&(I&E~). For brevity, in the rest of the proof the symbols “o(.)” and “N” will carry with them the uniformity features described here.

First, it follows from (3.2) and (3.4) that

(3.8)

a fact that will be applied a few times in this proof. By Boole’s inequality and suc- cessively applying the definition of a(.), taking into account the fact that the number of integers that separate Cl U el and Cj U Cj U C?j, I # j, is at least 2m,,

By stationarity, the definition of 6, and Boole’s inequality,

58 T. HsingIStochastic Processes and their Applications 60 (1995) 49-63

so that, by (3.4), we obtain

By (2.1) and (3.2),

m,n (P’K $ &,,I + 4mn 1) + 0,

and hence by (3.8),

lL4Ill = o( fphl).

I=1

We next analyze A,Q. First fix any A E (0,l). Since ( 1 - eti I< 1x1 A 2 for all x E R,

I1 - exp(iG, - r,)/~)&Eni) 1 <A;:)(A) +Az’(A), I=1 where

A;;)(A) = tAP h Enl , [ 1 I=1

and

Since

I(& - T,)/o,I > A, fj EnI . I=1 1

it follows from (3.8) that

A$( A) w tA fi P[E,,J I=1

Next,

A$)(4 .&‘[ 1 c X/G~ > W,i)E.i] j=l iGC,UC,UC?, i=l

~2e~[ ) c &/unl > A/k 1 Enj] fiPL%l+2k(k - l)a(m,). j=l iEC,U6,UtI I=1

T Hsing IStochastic Processes and their Applications 60 (1995) 4963 59

BY (3.81, 2W - l)a(m,) = o( n;=, PM). F or each j, by Boole’s inequality and

(3.4)

By (3.1), the right-hand side of the preceding inequality tends to 0 for each fixed A and therefore tends to 0 for some A = A,, + 0. Hence,

I.4~1 <&A,) +&A,) = o( fif’[&,). I=1

To handle A,,3, observe that the minimum distance between any member of D and any member of Uf=,(Cl U el) is m,. Successively applying the definition of E(.) and using Theorem 17.2.1 of Ibragimov and Linnik (1971) it follows from (3.8) that

IA,,31 <2k. 4. or(m,) = o( fif’[&t]). I=1

A,,4 is handled in the same way as A,,3, while it is straightforward to show that

A,,5 = o( nfz, P[&]). That An6 = o( nf=, P[&l]) follows from Theorem 1.1. Fi-

nally, that An7 = o can be proved by noticing that, with t = 0 in

Anjll d j < 6,

IAn G &Anj = 0( fiP[Enl]). (3.9) /=I I=1

Hence, (3.7) is shown, and (3.6) follows from (3.9) and (2.5). Thus, we have shown (2.4’) with 4, = i%Jn(ko, Eg) for any fixed b, ~0.

It is clear from the preceding proof that it is possible to pick k,, and E, which tend to cc and 0, respectively, slowly enough so that (2.4’) holds with .G?,, = @:n(kn,c,). Thus, what is left to show is that for any k, -+ cc and E, + 0, (2.3) holds with &,, = gn(kn,E,). Observe that it is sufficient to show the following:

lim P[k, E E for some B E ?4?:n,i] = 0, II-03

(3.10)

lim lim supP[*, E g for some b E 9&2(k)] = 0, k-m n-m

(3.11)

lim lim sup&?, E j for some B E &:,,3(1)] = 0, 1’00 n-cc

(3.12)

lim lim sup@, E i? for some B E !ZJn,4(E, s, Z)] = 0, E--+0 n-cc

s, 12 1, (3.13)

60 T. HsinglStochastic Processes and their Applications 60 (1995) 49-63

where

L&J = { B E L&+,, : B contains a pair of m,-clusters

with distance between them < 5m,),

&w = @ E ~:n,all : B contains at least k m,-clusters},

&,3(4 = @ E .%l,all : j contains a m,-cluster whose length is > I},

~n,&,S, 0 = { B E &,a,, : B contains s m,-clusters, which are at least 5m,

apart from one another, which have lengths at most 1,

and of which at least one violates (3.3)).

To show (3.10), simply note that

P[2, E B for some B E 4&t] <4nm, (p’[X, $ B,,,] + a(m,)),

which tends to 0 in a familiar way by (2.1) and (3.2). To show (3.11), apply Cheby shev’s inequality to get

P[J?, E L? for some B E 4&k)] <P [ kl(Xi $2 B,,)>k] <nP[Xl $! B,,]/k, i=i

from which (3.11) readily follows using (2.1). To show (3.12) note that

4 P[k, E B for some B E 4&Z)] 6nP[& +Z B,,x,U(& @ B,,,)]

i=l

and (3.12) follows readily from (2.1) and (2.2). Finally, for each g = (Bl,. . . , B,) E

.%A&, s, 0,

P[Jfn~j]<fiP n(XEBi)yn(&EBn,r) +(s-l)a(3m,), j=l iEC, iEt, 1

where Cl , . . . . C, are the m,-clusters of i. But, by (2.1) and (3.3),

(xi E Bi), n (4 E B,,T) iECT, 1 < constant s,

Since the number of such L? is bounded by (:)r’, we obtain

P[Z, E B for some I7 E .4:n,4(a,s, Z)] < z r’ 0 (

constant f + (s - l)r(3m,)).

Thus, (3.13) follows readily from this, and the proof of Theorem 2.1 is complete.

61 T. Hsing IStochastic Processes and their Applications 60 (1995) 4963

3.2. Proof of Theorem 2.2

Let JV be an infinite sequence of positive integers. It is clear that &/a,, NiL’, and N,‘“’ are individually tight (cf. Lemma 4.5 of Kallenberg, 1983), and hence jointly tight. Thus, there exists an infinite subsequence Jlr’ of JV along which (S,,/CJ~,N~~),N~(“‘) converges in distribution to some (Z,N CL) NC”)) in the product space R x A? x A. It , therefore suffices to show that 2 is independent of (NcL), NC”)), which follows from

P[Z<X,N(~)(CZ~,~~+~] = li,N(“)(ai,ai+i] = Ui, 1 <i<s]

= P[Z6x]P[N’L’(ai,ai+,] = li,N(“)(ai,ai+,] = Ui, 1 <i<s]

and which is, in turn, implied by

P[Sa/o, <x3 NiL)(ai, ai+,] = li, N,(“)(ai, ai+,] = Ui, 1 <i <s] (3.14)

= P[Sn/c, <X]P[N~L’(ai,ai+r] = Zi,N~“)(ai,ai+l] = pi, 1 <i<s] + O( 1)

for all positive integer S, positive real numbers 0 = al < u2 < . . . < us+1 < 03, and non-negative integers II,. . . , I, and ~1,. . . , u,. To show (3.14), let

Bn,i = {

{x 6 Iw : nF(x) E (@,Q+ll}, for l<i<s,

{X E R : nF(X) E (Ui_,y, Uifl _$I} for S + 1 < i < 2.7,

and

B n,&+, = {x E IR : nF(x) A n&x) > &+I}.

Now,

and, by (2.6),

Thus, with r = 2s + 1 and the B,i defined above, the conditions of (2.1) and (2.2) are satisfied, and hence the conclusions of Theorem 2.1 apply. In particular, let &-, be what is described there and let

4: = I

B E &“, : eI(Bj=B,i)=li for 1Qids and = Ui-_s for S + 1 <i62s .

j=l

62 T HsinglStochastic Processes and their Applications 60 (1995) 4963

Note that

(i,F td)) n(NiL’( u ai+,] = liyN,(“)(ai,a;+i] = Ui, 1 di6s i, >

Thus,

0 <P[Sn/‘o, <X,N,(L)(ai,ai+l] = li,Ni”)(Ui,ai+l] = Ui, 1 di<s]

-P s,/a, 6x, u (2, E B) BE.$’ ,i 1 <P (-)(2&B) . [ 1 BE&

With this and the fact that the events (2, E B),B E B,,, are disjoint, repeated application of Theorem 2.1 gives

P[S~/C, <X, N,(L)(Ui,ai+l] = li,N,(“)(ai,ai+l] = Ui, 1 <i<s]

= c P[S,/o, <x,2, E 81 + o( 1) BE.zJ:,

= P[S,/o, <xl c P[2, E E] + o( 1) BE.q

= P[S&, <x]P

= P[Sn/‘a, <x]P [Nn(L)(ai,ai+l] = li,Nn(“)(ai,ai+~] = Ui, 1 <ids] + O(l),

where the exact expressions for the o( 1) terms differ from line to line. This shows (3.14) and completes the proof.

Acknowledgements

I am grateful to the referees for their valuable comments.

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